A Full Characterization of Bertrand Numeration Systems

Abstract

We consider Cantor real numeration system as a frame in which every nonnegative real number has a positional representation. The system is defined using a bi-infinite sequence B = (βn) n∈Z of real numbers greater than one. We introduce the set of B-integers and code the sequence of gaps between consecutive B-integers by a symbolic sequence in general over the alphabet N. We show that this sequence is S-adic. We focus on alternate base systems, where the sequence B of bases is periodic and characterize alternate bases B, in which B-integers can be coded using a symbolic sequence vB over a finite alphabet. With these so-called Parry alternate bases we associate some substitutions and show that vB is a fixed point of their composition. The paper generalizes results of Fabre and Burdík et al. obtained for the Rényi numerations systems, i.e., in the case when the Cantor base B is a constant sequence.

String Attractors of Fixed Points of k-Bonacci-Like Morphisms

“…Note that, if the numeration system S c satisfies the greedy condition, this result follows from the characterization of numeration systems in terms of dynamical systems given by Bertrand-Mathis [3,5]. However, even though the function rep Sc is obtained using the word-greedy factorization of prefixes of u, the numeration system S c is not necessarily greedy as the following example shows.…”

“…In addition, some of these morphisms have already been studied in relation to numeration systems, in [13] for example. Indeed, if c is some β-representation of 1 for a simple Parry number β, using the terminology of [5], we can canonically associate a numeration system that is greedy and, in this case, corresponds to the sequence (|µ n c (0)|) n∈N of lengths of iterations of µ c on 0 [3]. Under some conditions on the parameters, we show that the prefixes of the fixed point admit string attractors of size at most k + 1 described using the associated numeration system.…”

Alternate Base Numeration Systems

“…The so-obtained expansion d 0 • • • d ℓ−1 is called the U -expansion of x. Similarly, the literature about U -expansions of nonnegative integers is vast, see [4,5,12,13,18,21,26] for the most topic-related ones.…”